JGSP 9 (2007) 9–31
LOCAL CALCULATION OF HOFER’S METRIC AND
APPLICATIONS TO BEAM PHYSICS
BELA ERDELYI
Communicated by Izu Vaisman
Abstract. Hofer’s metric is a very interesting way of measuring distances between compactly supported Hamiltonian symplectic maps. Unfortunately, it is not
known yet how to compute it in general, for example for symplectic maps far
away from each other. It is known that Hofer’s metric is locally flat, and it can be
computed by the so-called oscillation norm of the difference between the Poincare
generating functions of symplectic maps close to identity. It is shown here that the
same result holds for arbitrary extended generating function types and symplectic
maps, as long as the respective generating functions are well defined for the given
symplectic maps. This result plays a crucial role is formulating and solving the
optimal symplectic approximation problem in Hamiltonian nonlinear dynamics.
Applications to beam physics are oulined.
1. Introduction
Often, in Hamiltonian dynamics, it is necessary to study the quality of the approximation of the real dynamics by some approximate numerical or analytical
method, since the equation of the motion cannot be solved analytically in closed
form. Lately, the method of choice is the symplectic integration method, which
is a category of the so-called geometric integrators [10]. For very complicated
weakly nonlinear Hamiltonian systems, as for example large particle accelerators,
a variant of the symplectic integration methods is utilized, called symplectification of one-turn maps. To this end, an approximate functional relationship is calculated for trajectories of particles one turn around the accelerator, and the longterm behavior of the beam is obtained by iterating this map. The one-turn map’s
approximation is not exactly symplectic, and the method that transforms this map
into a symplectic map is called symplectification [19]. The study of the symplectification in an optimal way lead to the consideration of Hofer’s metric [11]
for the formulation of the optimal symplectification conditions [6]. However, the
same theory may be applied to symplectic integrators in the traditional sense. The
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